The Pythagorean theorem

1. The Pythagorean theorem Although Pythagoras is credited with the famous theorem, it is likely that the Babylonians knew the result for certain specific triangles at least a millennium earlier than Pythagoras. It is not known how the Greeks originally demonstrated the proof of the Pythagorean Theorem. The Pythagorean theorem

2. The Pythagorean theorem Right triangle "In any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs." C 90° hypotenuse A B The Pythagorean theorem

3. The Pythagorean theorem The sum of the areas of the two squares on the legs equals the area of the square on the hypotenuse. The Pythagorean theorem

4. Demonstrate the Pythagorean Theorem Many different proofs exist for this most fundamental of all geometric theorems 16(42) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 9(32) 1 2 3 4 5 6 7 8 9 C 90° 25=9+16 hypotenuse Right triangle A B 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 The sum of the areas of the two squares on the legs equals the area of the square on the hypotenuse. 25(52) The Pythagorean theorem

5. Several beautiful and intuitive proofs by shearing exist 3 4 5 C 1 2 B A 5 3 The sum of the areas of the two squares on the legs equals the area of the square on the hypotenuse. 4 1 2 The Pythagorean theorem

6. "The area of the square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides." The Pythagorean theorem

7. The Indian mathematician Bhaskara constructed a proof using the above figure, and another beautiful dissection proof is shown below The sum of the areas of the two squares on the legs equals the area of the square on the hypotenuse. The Pythagorean theorem

8. Pythagorean Triples The Pythagorean theorem

9. Pythagorean Triples There are certain sets of numbers that have a very special property.  Not only do these numbers satisfy the Pythagorean Theorem, but any multiples of these numbers also satisfy the Pythagorean Theorem. For example: the numbers 3, 4, and 5 satisfy the Pythagorean Theorem.  If you multiply all three numbers by 2  (6, 8, and 10), these new numbers ALSO satisfy the Pythagorean theorem.   The special sets of numbers that possess this property are called Pythagorean Triples. The most common Pythagorean Triples are: 3, 4, 5 5, 12, 13 8, 15, 17 The Pythagorean theorem

10. The formula that will generate all Pythagorean triples first appeared in Book X of Euclid's Elements: where n and m are positive integers of opposite parity and m>n. The Pythagorean theorem

11. A triangle has sides 6, 7 and 10. Is it a right triangle? The longest side MUST be the hypotenuse, so c = 10.  Now, check to see if the Pythagorean Theorem is true. Since the Pythagorean Theorem is NOT true, this triangle is NOT a right triangle. The Pythagorean theorem"In any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs." The Pythagorean theorem

12. The Distance Formula The Pythagorean theorem

13. The distance between points P1 and P2 with coordinates (x1, y1) and (x2,y2) in a given coordinate system is given by the following distance formula: The Pythagorean theorem

14. To see this, let Q be the point where the vertical line trough P2 intersects the horizontal line trough P1. • The x coordinate of Q is x2 , the same as that of P2. • The y coordinate of Q is y1 , the same as that of P1. • By the Pythagorean theorem . The Pythagorean theorem

15. If H1 and H2are the projection of P1 and P2on the x axis, the segments P1Q and H1H2 are opposite sides of a rectangle, so that But so Similarly, The Pythagorean theorem

16. Hence Taking square roots, we obtain the distance formula: The Pythagorean theorem

17. EXAMPLE The distance between points A(2,5) and B(5,9) is The Pythagorean theorem